A monolithic ALE Newton–Krylov solver with Multigrid-Richardson–Schwarz preconditioning for incompressible Fluid-Structure Interaction

Abstract

In this paper we study a monolithic Newton-Krylov solver with exact Jacobian for the solution of incompressible FSI problems. A main focus of this work is on the use of geometric multigrid preconditioners with modified Richardson smoothers preconditioned by an additive Schwarz algorithm. The definition of the subdomains in the Schwarz smoother is driven by the natural splitting between fluid and solid. The monolithic approach guarantees the automatic satisfaction of the stress balance and the kinematic conditions across the fluid-solid interface. The enforcement of the incompressibility conditions both for the fluid and for the solid parts is taken care of by using inf-sup stable finite element pairs without stabilization terms. A suitable Arbitrary Lagrangian Eulerian (ALE) operator is chosen in order to avoid mesh entanglement while solving for large displacements of the moving fluid domain. Numerical results of two and three-dimensional benchmark tests with Newtonian fluids and nonlinear hyperelastic solids show a robust performance of our fully incompressible solver especially for the more challenging direct-to-steady-state problems.

A field‐split preconditioning technique for fluid‐structure interaction problems with applications in biomechanics

“…In the following, we focus on the analysis of the momentum balance, and we briefly describe the role of the function in Equation (11). For a full description of the PDE system (10)‐(14), please refer to Aulisa et al Equation (12) describes, in a monolithic form, the solid and fluid momenta, which are also referred to as the incompressible nonlinear elasticity equation and the Navier‐Stokes equation, respectively. The symbols ρ f and ρ s denote the mass densities for the fluid and solid part, respectively, whereas f f and f s indicate the body force densities.…”

“…Recent works showed that the above equation ordering is ill‐conditioned for steady‐state problems, and it becomes unstable for time‐dependent problems as the time step increases. Such a behavior can be easily explained by looking at the diagonal terms and in the first and third rows of (26).…”

“…To analyze the FS preconditioner for the level subsolvers, we compare its performance with the preconditioner in Aulisa et al, which is of pure domain decomposition type. In Aulisa et al and Calandrini et al, the Krylov solver in Aulisa et al has been applied to solve FSI problems with biomedical applications, namely, it has been used for stented cerebral aneurysm simulations in Aulisa et al and for magnetic drug targeting (MDT) procedures in Calandrini et al We intend to show that an FS approach is more suitable for biomedical FSI problems, where a Krylov solver preconditioned with a multigrid method is adopted.…”

Magnetic drug targeting simulations in blood flows with fluid‐structure interaction

“…The solid motion is described in a Lagrangian way, while the fluid is observed in Eulerian fashion. The deformation of the fluid domain is taken into account according to an arbitrary Lagrangian‐Eulerian (ALE) approach, which is one of the most popular techniques in the FSI community . To solve the FSI system, we use a monolithic Newton‐Krylov solver preconditioned by a geometric multigrid algorithm .…”

Fluid-Structure Simulations and Benchmarking of Artery Aneurysms Under Pulsatile Blood Flow

“…We describe the solid motion in a Lagrangian way, while the fluid is observed in Eulerian fashion. The deformation of the fluid domain is taken into account according to an Arbitrary Lagrangian Eulerian (ALE) approach, which is one of the most popular techniques in the FSI community ( [4], [13], [23], [29]). To linearize the FSI system, Newton linearization is performed, therefore the evaluation of the Jacobian associated to the fluid-solid coupled state equations is required.…”